3

This question is inspired by this one which topic was finding all the rational solutions of the Pell’s equation $x^2-dy^2=1$.

The answers show that these solutions can be described, for instance, by the parameterization

$$x=\dfrac{t^2+d}{t^2-d},y=\frac{2t}{t^2-d},(t\in \mathbb Q,t^2\neq d).$$

This also gives a parameterization of the norm 1 subgroup of the number field $\mathbb{Q}[\sqrt{d}]$. I was wondering if the same could be done with the norm 1 subgroup of $\mathbb{Q}[\sqrt[3]{d}]$. For instance, if $d=2$, I’m looking for all the rational solutions of the norm equation

$$a^3+2b^3+4c^3-6abc=1.$$

One of the answers on the other question suggests using Hilbert’s Theorem 90 for this kind of problems. However, I haven’t learned the proof of this theorem yet, as I’m still taking an introductory number theory course and don’t know much yet about Galois theory. So I was wondering if there was another method (either in general or that could be applied to this particular problem) useful here or if someone could explain to me how Hilbert’s Theorem 90 applies to this particular situation (what would the homomorphism $\sigma$ be here?).

dahemar
  • 1,774
  • 3
    https://link.springer.com/chapter/10.1007/978-3-030-01404-9_9 (which is on sci-hub) is exactly about that. The trick seems to treat directly the norm 1 problem for the generic cubic extension $\Bbb{Q}(u,v)[x]/(x^3+ux+v)\ /\Bbb{Q}(u,v)$. – reuns Feb 17 '23 at 23:29
  • 1
    Be careful about hoping too many geometric figures have a rational parametrization. That a smooth curve has a rational parametrization is related to whether or not the curve has genus 0. Elliptic curves, for instance, have genus 1 and their points do not admit a rational parametrization. And $\mathbf Q(\sqrt[3]{2})$ is not a Galois extension of $\mathbf Q$ (all quadratic extensions of $\mathbf Q$ are Galois extensions) and in fact the only field automorphism of $\mathbf Q(\sqrt[3]{2})$ is the identity. So you can't apply Theorem 90 to the cubic setting in the way you are trying to do. – KCd Feb 18 '23 at 04:01
  • @KCd Thank you, that information was helpful as I’m still learning about all of this. There’s no “equivalent” of Theorem 90 in this setting, right? I mean a general method to solve this kind of problems (that could be applied, for instance, as you were saying, to curves of genus $0$)? – dahemar Feb 18 '23 at 10:00
  • The norm-$1$ subgroup of a number field is simply the unit group of the number field, or an index-$2$ subgroup thereof. Dirichlet's unit theorem tells you its structure, and is a good starting point for parametrizing it. – Servaes Feb 24 '23 at 00:39

0 Answers0